Representation Theory of Semigroups

给定集合上的任何数学结构，该集合到其自身的结构保持映射的集合是称为半群的抽象代数对象的一个例子。因此，半群弥漫在数学中。另一方面，给定一个抽象定义的半群，什么时候才能将其表示为具有数学结构的映射的半群？我们将在我们的三个机构中进行三个主要的研究方向：动作的自动性(St.Andrews)，动作和部分动作在半群结构和分类中的使用(York)，以及逆半群的动作(Heriot-Watt)。然而，正如在我们的支持案例中所解释的那样，这些链之间存在许多相互作用。我们的目标是收集现有的材料，将其置于一个共同的框架中，并利用我们的联合专业知识来解决一些悬而未决的问题。这将与来自全球的该领域的领先研究人员一起以合作的方式完成。使用自动机研究诸如半群之类的代数，在代数和理论计算机科学之间建立了一座桥梁，使我们能够使用有限状态自动机来定义无限代数。自动机群和半群现在被广泛研究，但尽管动作的概念被严重依赖，自动动作的研究(由Dombi引入)还处于初级阶段。自动群的几何结果，如与同行者性质的等价性，不会延续到半群。我们的目标是使用自动作用来开发自动半群的新概念，这将在一定程度上弥合这些差距。我们将考虑随后的性质，建立新的不可判定结果和计算半群的算法。逆半群是构成几何中描述局部结构的基础的变换伪群的代数版本。对于每个逆半群，我们可以结合一个拓扑群，由这些拓扑群可以构造C~*-代数。因此，逆半群、拓扑群和C*-代数是密切相关的，构成了非交换几何的一个重要组成部分。其指导思想是逆半群的表示理论为研究部分对称性提供了一个统一的框架。这可以被看作是群的表示理论为研究对称性提供了统一的框架的方式的深远推广。例如，逆半群可以与非周期平铺联系在一起，由此产生的群胚是Stone对偶的非对易推广的一部分。此外，平铺半群的表示被认为控制了群胚的结构，从而控制了相关的C*-代数。集合的部分映射(粗略地说，不是处处定义的映射)何时可以(以适当的方式)扩展到全局映射的问题，是代数和模型理论的核心问题。在许多结构定理中，半群在集合和序结构上的部分作用被隐含地使用，但尚未被利用。我们将研究半群在有结构的集合上的部分作用何时可以‘全球化’，以及在有限的情况下，这个问题是否可判定。我们相信这是解决悬而未决的问题的关键，例如，每个有限逆半群都有有限F-逆覆盖吗？我们还将结合我们的专业知识，试图破解经典行动理论中长期悬而未决的问题。该项目将涉及5名永久研究人员：三名提名者，一名研究助理和一名博士生。它还将涉及一系列研究访问和与该领域领先专家的合作。我们将组织一次早期研讨会，以开始合作进程，并确保我们采取包容的方法进行研究。

Given any mathematical structure on a set, the collection of structure-preserving maps of the set to itself is an example of an abstract algebraic `object' called a semigroup. Thus, semigroups pervade mathematics. On the other hand, given an abstractly defined semigroup, when can it be represented as a semigroup of maps of a mathematical structure? If so, we say that it is represented by actions.Three main strands of research will be pursued in our three institutions: automaticity of actions (St. Andrews), the use of actions and partial actions in the structure and classification of semigroups (York), and actions of inverse semigroups (Heriot-Watt). However, as explained in our Case for Support, there are many interactions between these strands. We aim to draw together existing material, place it in a common framework, and use our combined expertise to solve a number of outstanding problems. This will be done in a collaborative way, together with leading researchers in the area from across the globe. Studying algebras such as semigroups using automata builds a bridge between algebra and theoretical computer science, allowing us to define infinite algebras using finite state automata. Automatic groups and semigroups are now widely studied, but although the notion of action is heavily relied upon, the study of automatic actions (introduced by Dombi) is in its infancy. Geometric results for automatic groups, such as the equivalence to the fellow traveller property, do not carry over for semigroups. We aim to use automatic actions to develop new notions of automatic semigroup, which will go some way to bridging these gaps. We will consider subsequent properties, establishing new undecidability results and algorithms to calculate semigroups. Inverse semigroups are the algebraic versions of the pseudogroups of transformations that form the foundation for describing local structures in geometry. With each inverse semigroup one can asssociate an etale topological groupoid and from such groupoids one can construct C*-algebras. Thus inverse semigroups, etale topological groupoids, and C*-algebras are closely related, forming an important ingredient in non-commutative geometry. The guiding idea is that the representation theory of inverse semigroups provides a unifying framework for studying partial symmetries. This can be seen as a far-reaching generalization of the way in which the representation theory of groups provides a unifying framework for studying symmetries. For example, inverse semigroups can be associated with aperiodic tilings, and the groupoids that result form part of a non-commutative generalization of Stone duality. Furthermore, the representations of the tiling semigroups are known to control the structure of the groupoids, and hence the associated C*-algebras.The question of when a partial map of a set (roughly speaking, a map not everywhere defined) can be extended (in a suitable way) to a global map, is central to aspects of algebra and model theory. Partial actions of semigroups on sets and ordered structures are used implicitly in many structure theorems, but yet have not been exploited. We will investigate when the partial action of a semigroup on a set with structure can be `globalised', and, in the finite case, whether this question is decidable. We believe this is the key to solving outstanding questions, such as, does every finite inverse semigroup has a finite F-inverse cover? We will also use our combined expertise to try to crack long unsolved questions from the classical theory of actions.The project will involve 5 permanent researchers: the three proposers, a Research Assistant and a PhD student. It will also involve a string of research visits and collaborations with leading experts in the field. We will organise an early Workshop to begin the collaborative process and to ensure we take an inclusive approach to our research.